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Homefield adjunction

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Let $K$ be a field and $E$ an extension field of $K$. If $\alpha\in E$, then the smallest subfield of $E$, that contains $K$ and
$\alpha$, is denoted by $K(\alpha)$. We say that $K(\alpha)$ is
obtained from the field $K$ by adjoining the element $\alpha$
to $K$ via field adjunction.

Theorem. $K(\alpha)$ is identical with the quotient field $Q$ of $K[\alpha]$.

Proof. (1) Because $K[\alpha]$ is an integral domain (as a subring of the field $E$), all possible quotients of the elements of $K[\alpha]$ belong to $E$. So we have

$K\cup\{\alpha\}\subseteq K[\alpha]\subseteq Q\subseteq E,$ |

and because $K(\alpha)$ was the smallest, then $K(\alpha)\subseteq Q.$

(2) $K(\alpha)$ is a subring of $E$ containing $K$ and $\alpha$, therefore also the whole ring $K[\alpha]$, that is, $K[\alpha]\subseteq K(\alpha)$. And because $K(\alpha)$ is a field, it must contain all possible quotients of the elements of $K[\alpha]$, i.e., $Q\subseteq K(\alpha)$.

In addition to the adjunction of one single element, we can adjoin to $K$ an arbitrary subset $S$ of $E$: the resulting field $K(S)$ is the smallest of such subfields of $E$, i.e. the intersection^{} of such subfields of $E$, that contain both $K$ and $S$. We say that $K(S)$ is obtained from $K$ by adjoining the set $S$ to it. Naturally,

$K\subseteq K(S)\subseteq E.$ |

The field $K(S)$ contains all elements of $K$ and $S$, and being a field, also all such elements that can be formed via addition, subtraction, multiplication and division from the elements of $K$ and $S$. But such elements constitute a field, which therefore must be equal with $K(S)$. Accordingly, we have the

Theorem. $K(S)$ constitutes of all rational expressions formed of the elements of the field $K$ with the elements of the set $S$.

Notes.

1. $K(S)$ is the union of all fields $K(T)$ where $T$ is a finite subset of $S$.

2. $K(S_{1}\cup S_{2})=K(S_{1})(S_{2})$.

3. If, especially, $S$ also is a subfield of $E$, then one may denote $K(S)=KS$.

# References

- 1
B. L. van der Waerden: Algebra
^{}. Erster Teil. Siebte Auflage der Modernen Algebra. Springer-Verlag; Berlin, Heidelberg, New York (1966).

## Mathematics Subject Classification

12F99*no label found*

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